A12479. Serval and Bonus Problem

Getting closer and closer to a mathematician, Serval becomes a university student on math major in Japari University. On the Calculus class, his teacher taught him how to calculate the expected length of a random subsegment of a given segment. Then he left a bonus problem as homework, with the award of a garage kit from IOI. The bonus is to extend this problem to the general case as follows.

You are given a segment with length $l$ . We randomly choose $n$ segments by choosing two points (maybe with non-integer coordinates) from the given segment equiprobably and the interval between the two points forms a segment. You are given the number of random segments $n$ , and another integer $k$ . The $2n$ endpoints of the chosen segments split the segment into $(2n+1)$ intervals. Your task is to calculate the expected total length of those intervals that are covered by at least $k$ segments of the $n$ random segments.

You should find the answer modulo $998244353$ .

First line contains three space-separated positive integers $n$ , $k$ and $l$ ( $1\leq k \leq n \leq 2000$ , $1\leq l\leq 10^9$ ).

Output one integer — the expected total length of all the intervals covered by at least $k$ segments of the $n$ random segments modulo $998244353$ .

Formally, let $M = 998244353$ . It can be shown that the answer can be expressed as an irreducible fraction $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{M}$ . Output the integer equal to $p \cdot q^{-1} \bmod M$ . In other words, output such an integer $x$ that $0 \le x < M$ and $x \cdot q \equiv p \pmod{M}$ .

In the first example, the expected total length is $\int_0^1 \int_0^1 |x-y| \,\mathrm{d}x\,\mathrm{d}y = {1\over 3}$ , and $3^{-1}$ modulo $998244353$ is $332748118$ .